Quantum Man: Richard Feynman's Life in Science (6 page)

While Melville Feynman was concerned about his son’s career direction, Richard’s mother, Lucille, was equally concerned about his personal one. She surely loved Arline, but she wrote to him, late in his graduate career, like many a Jewish mother, concerned that Arline would be a drag on his ability to work and gain a job and on his finances. Arline’s illness would require special care, and time and money, and Lucille was worried that Richard didn’t have enough of any of these things.

Richard responded, within weeks of receiving his
P
hD and marrying Arline, in June of
1
94
2
, remarkably dispassionately:

I’m not dopey enough to tie up my whole life in the future because of some promise I made in the past—under different circumstances. . . . I want to marry Arline because I love her—which means I want to take care of her. That is all there is to it. . . .

I have, however, other desires and aims in the world. One of them is to contribute as much to physics as I can. This is, in my mind, of even more importance than my love for Arline.

It is therefore especially fortunate that, as I can see (guess) my getting married will interfere very slightly, if at all with my main job in life. I am quite sure I can do both at once. (There is even the possibility that the consequent happiness of being married—and the constant encouragement and sympathy of my wife will aid in my endeavor—but actually in the past my love hasn’t affected my physics much, and I don’t really suppose it will be too great an assistance in the future.)

Since I feel I can carry on my main job, and still enjoy the luxury of taking care of someone I love—I intend to be married shortly.

Whether or not his love affected his physics, Arline had clearly reinforced his determination to follow his ideas wherever they might lead. She had helped ensure his intellectual integrity, and if the words in his letter seem somewhat cold and dispassionate, Arline might have been encouraged by them had she ever read them, because they reflected the kind of rational thinking she so wanted to foster in the man she so loved and admired.

She might have been equally moved by a heart-wrenching event that happened much later, on the dark day of her death, June 16, 1945, six weeks before the atomic bomb Richard had worked to build was exploded over Hiroshima. After she breathed her last breath in the hospital room, he kissed her, and the nurse recorded the time of death as 9:21 p.m. He later discovered that the clock by her bedside had stopped at precisely the same time. A less rational mind might have found this cause for spiritual wonder or enlightenment—the kind of phenomena that makes people believe in a higher cosmic intelligence. But Feynman knew the clock was fragile. He had fixed it several times and he reasoned that the nurse must have picked it up and disturbed it to check the time of Arline’s death. He would display the same kind of intellectual focus and determination to continue down a road he began in 1941, one that would ultimately, profoundly, and irrevocably change the way we think about the world.

T
HE WRITER LOUISE
Bogan once said, “The initial mystery that attends any journey is: how did the traveler reach his starting point in the first place?” For Feynman’s journey, like many epic voyages, the beginning was simple enough. He and Wheeler had completed their work demonstrating that classical electromagnetism could be cast in a form that involved only direct interactions, albeit forward and backward in time, between different charged particles. In so doing, one could obviate the problem of the infinite self-energy of any individual charged particle. The next challenge was to see if this theory could be brought into accord with quantum mechanics, and possibly resolve the thornier mathematical problems that resulted in a quantum theory of electromagnetism.

The only problem was that their rather exotic theory— which was rigged with interactions at different times and places in order to be equivalent in its predictions with the results of classical electromagnetism, and which had electric and magnetic fields that transmitted these interactions—required a mathematical form that quantum mechanics couldn’t handle at the time. The problem originated because of the interactions between particles at different times, or as Feynman later put it, “The path of one particle at a given time is affected by the path of another at a different time. If you try to describe, therefore, things . . . telling what the present conditions of the particles are, and how these present conditions will affect the future—you see it is impossible with particles alone, because something the particles did in the past is going to affect the future.” Up to that point quantum mechanics was based on a simple principle. If we somehow knew or were told the quantum state of a system at one time, the equations of quantum mechanics allowed us to determine precisely the subsequent dynamical evolution of the system. Of course, knowing precisely the dynamical evolution of the system is not the same as predicting exactly what we would subsequently measure. The dynamical evolution of a quantum system involves determining exactly, not the final state of the system, but rather a set of probabilities that tells us what the likelihood is the system being measured will be in some specific state at a later time.

The problem is that electrodynamics as formulated by Feynman and Wheeler required knowing the positions and motion of many other particles at many different times in order to determine the state of any one given particle at any given time. In such a case, the standard quantum methods for determining the subsequent dynamical evolution of this particle failed.

Feynman had succeeded during the fall and winter of 1941–42 in formulating their theory in a host of different, if mathematically equivalent, ways. During the process he had discovered that he could rewrite the theory completely in terms of the very principle that he had so renounced while an undergraduate.

Remember that Feynman had learned in high school that there was a formulation of the laws of motion which was based not on what was happening at a single time, but what happened at all times: the formalism of Lagrange and his principle of least action.

Recall also that the least action principle tells us that in order to determine the actual classical trajectory of a particle, we can consider all possible paths of the particle between its beginning and end points and then determine which one has the smallest average value for the action (defined as the differences between two different parts of the total energy of the particle—the so-called kinetic and potential energies—appropriately summed over each path). This was the principle that was too elegant for Feynman, who preferred to calculate trajectories by considering forces at every point and using Newton’s laws. The idea that he had to worry about the entire path of a particle in order to calculate its behavior at any point seemed unphysical to Feynman at the time.

But Feynman the graduate student had discovered that his theory with Wheeler could be recast completely in terms of an action principle, described purely by the trajectories of charged particles over time, with no need to consider electric and magnetic fields. In retrospect it seems clear why such formalism, which focused on the paths of particles, was appropriate to describe the Feynman-Wheeler theory. After all, such paths are essentially what defined their theory, which depended completely on the interactions of particles moving along different trajectories in time. Therefore, to build a quantum theory, Feynman decided he would need to figure out how to do quantum mechanics for a system like the one he and Wheeler were considering, whose classical dynamics could be determined by such an action principle, but
not
by more conventional methods.

Physics, or at least the physics that Feynman and Wheeler were imagining, had driven Feynman to a place he never would have expected to be a half-dozen years earlier. The transformation in his thinking following his intensive efforts to explore their new theory had been dramatic. He was now convinced that focusing on events at a fixed time was not the way to think, and that the action principle, based on exploring complete trajectories through space and time, was. As he later wrote, “We have in [the action principle] a thing that describes the character of the path throughout all of space and time. The behavior of nature is determined by saying her whole-space-time path has a certain character.”
But how could this principle be translated into quantum mechanics, which thus far depended so crucially on defining a system at one time in order to calculate what would happen at later times? For Feynman, the key to the answer came from a fortuitous beer party in Princeton. But to appreciate this key, we first have to make a short detour to revisit our picture of the mysterious quantum world that Feynman was about to change.

CHAPTER
4

Alice in Quantumland

The Universe is not only queerer than we suppose, but queerer than we
can
suppose.

J. B. S. H
ALDANE
, 1924

W
hile the distinguished British scientist J. B. S. Haldane was a biologist and not a physicist, his statement about the universe could not be more apt, at least to the realm of quantum mechanics that Richard Feynman was about to conquer. For, as we have seen, at the small scales where quantum mechanical effects become significant, particles can appear to be in many different places at once, while also doing many different things at the same time in each place.

The mathematical quantity that can account for all of this apparent lunacy is the function discovered by the famous Austrian physicist Erwin Schrödinger, who derived what became the conventional understanding of quantum mechanics during a busy two-week period in which he too was doing many different things at the same time—in the midst of trysts with perhaps two different women while holed up in a mountain chalet. It probably was the perfect atmosphere to imagine a world where all of the classical rules of behavior would ultimately be broken.

This function of Schrödinger’s is called the
wave function
of an object, and it accounts for the mysterious fact, at the heart of quantum mechanics, that all particles behave in some sense like waves, and all waves behave in some sense like particles—the difference between a particle and a wave being that a particle is located at a specific point, whereas a wave is spread out over some region.

So, if a particle, which isn’t spread out, is to be described by something that behaves like a wave and is spread out, the wave function must accommodate this fact. As Max Born later demonstrated, this was possible if the wave function, which itself might behave like a wave, did not describe the particle itself but rather the
probability
of finding the particle at any given place in space at a specific time. If the wave function, and hence the probability of finding a particle, is nonzero at many different places, then the particle acts like it is in many different places at any one time.

So far so good, even if the notion itself seems crazy. But there is one more crucial bit of craziness at the heart of quantum mechanics, and I should stress that physicists do not have a fundamental understanding of why nature behaves this way, except to say that it does. If the laws of quantum physics determine the behavior of the wave function, then physics tells us that given the wave function of a particle at one time, quantum mechanics in principle allows us to calculate, in a completely deterministic way, the wave function of the particle at a later time. Up to this point it is just like Newton’s laws, which tell us how the classical motion of a baseball evolves in time, or Maxwell’s equations, which tell us how electromagnetic waves evolve in time. The difference is that in quantum mechanics the quantity that evolves in time in a deterministic manner is not directly observable, but rather is a set of probabilities for making certain observations, in this case for determining the particle to be at a certain place at a certain time.

This is strange enough, but it further turns out that the wave function itself does not directly describe the probability of finding a particle at a given place at some time. Instead it is the
square
of the wave function that gives the probabilities. This one fact is responsible for all of the strangeness of the quantum mechanical world because it explains why particles can behave precisely as waves, as I will describe now.

First, note that the probabilities of things we measure must be positive (we would never say that there is a probability of minus 1 percent of finding something) and the square of a quantity is also always positive, so quantum mechanics predicts positive probabilities—which is a good thing. But it also implies that the wave function itself can be either positive or negative, since, say, −½ and +½ both yield the same number (+¼) when squared.

If it were the wave function that described the probability of finding some particle at some location
x
, then if I had two identical particles, the probability of finding either particle at location
x
would be the sum of the two individual (and each necessarily positive) wave functions. However, because the square of the wave function is what determines the probability of finding particles, and because the square of the sum of two numbers is
not
equal to the sum of the individual squares, things can get much more interesting in quantum mechanics.

Let’s say the value of the wave function that corresponds to finding particle
A
at position
x
is P1, and the value of the wave function that corresponds to finding particle
B
at position
x
is P2, then quantum mechanics tells us that the probability of finding either particle
A
or
B
at position
x
is now (P1 + P2)
2
. Let’s say P1 = ½ and P2 = −½. Then if we only had one particle, say particle
A
, the probability of finding it at position
x
would be (½)
2
= ¼. Similarly the probability of finding particle
B
at position
x
would be (−½)
2
= ¼. However, if there are two particles, the probability of finding
either
particle at position
x
is ((½) + (−½))
2
= 0.

This phenomenon, which on the surface seems ridiculous, is in fact familiar for waves, say, sound waves. Such waves can
interfere
with each other so that, for example, waves on a string can interfere and produce locations on the string, called
nodes
, that do not move at all. Similarly, if sound waves are coming from two different speakers in a room, we might find, if we were to walk around the room, certain locations where the waves cancel each other out, or, as physicists say,
negatively interfere
with each other. (Acoustic experts design concert halls so that hopefully there are no such “dead spots.”)

What quantum mechanics, with probabilities being determined by the square of the wave function, tells us is that particles too can
interfere
with each other, so that if there are two particles in a box, the probability of finding either of them at a given location can end up being less than the probability of finding one where only a single particle is in the box.

When waves interfere, it is the height, or
amplitude
, of the resulting wave that is affected by this interference, and the amplitude can be positive or negative depending on whether one is at a peak or a trough in the wave. So another name for the wave function of a particle is its
probability
amplitude
, which can be positive or negative.

And just as for regular amplitudes for sound waves, separate probability amplitudes for different particles can cancel each other out.

It is precisely this mathematics that is behind the behavior of electrons shot at a scintillating screen, as described in chapter 2. Here we find that an electron can actually interfere with itself because electrons have a nonzero probability of being in many different places at any one time.

Let’s first think about how to calculate probabilities in a sensible, classical world. Consider choosing to travel from town
a
to town
c
by taking some specific route through town
b
. Let the probability of choosing some route from
a
to
b
be given by P(
ab
), and then the probability of choosing some specific route from
b
to
c
be P(
bc
). Then, if we assume that what happens at
b
is completely independent of what happens at
a
and
c
, the probability of traveling from
a
to
c
along a specific route that goes through town
b
is simply given by the product of the two probabilities, P(
abc
) = P(
ab
) × P(
bc
). For example, say there is a 50 percent chance of taking some route from
a
to
b
, and then a 50 percent chance of taking some route from
b
to
c
. Then if we were to send four cars out, two will make it to
b
on the chosen route, and of those two, one will take the next chosen route from
b
to
c
. Thus there is a 25 percent (.5 × .5) probability of taking the required route all the way from beginning to end.

Now, say we don’t care which particular point
b
is visited between
a
and
c
. Then the probability of traveling from
a
to
c
, given by P(
ac
), is simply the sum of the probabilities P(
abc
) of choosing to go through
any
point
b
between
a
to
c
.

The reason this makes sense is that classically if we are going from
a
to
c
, and
b
represents the totality of different towns we can cross through, say, halfway from both
a
and
c
, then we have to go through one of them during our journey (see figure).

(Since this picture is reminiscent of the earlier pictures of light rays, then we could say that if the example in question involved light rays going from
a
to
c
, then we could use the principle of least time to determine that the probability of going through one of the routes, that of least time, was 100 percent, and the probability of taking any other route was zero.)

The problem is that things don’t work this way in quantum mechanics. Because probabilities are determined by the squares of probability amplitudes to go from one place to another, the probability to go from
a
to
c
is
not
given by the sum of the probabilities to go from
a
to
c
via any definite intermediate point
b
. This is because in quantum mechanics it is the separate probability
amplitudes
for each part of the route that multiply and not the probabilities themselves. Thus, the probability amplitude to go from
a
to
c
through some definite point
b
is given by multiplying the probability amplitude to go from
a
to
b
times the probability amplitude to go from
b
to
c
.

If we don’t specify which point
b
to travel through, the probability amplitude to go from
a
to
c
is again given by the sum of the product of probability amplitudes to go from
a
to
b
and from
b
to
c
, for all possible
b
’s. But this means that actual probability is now given as the
square
of the sum of these products. Since some terms in the sum can be negative, the crazy quantum behavior I discussed in chapter 2 for electrons hitting a screen can occur. Namely, if we don’t measure which of two points, say
b
and
b
'
, a particle traverses as it travels through one of two slits between
a
and
c
, then the probability of arriving at point
c
on the screen is determined by the sum of the squares of the probability amplitudes for the two different allowed paths. If we do measure which point,
b
or
b
'
, the particle traverses in between
a
and
c
, then the probability is simply the square of the probability amplitude for a single path. In the case of many electrons shot one at a time, the final pattern on the screen in the former case will be determined by adding the squares of the sum of probability amplitudes for each of the two possible paths for each particle, while in the latter case it will be determined by adding the squares of the probability amplitudes for each path separately taken by each electron. Again, because the square of a sum of numbers is different from the sum of squares of these numbers, the former probability can differ dramatically from the latter. And as we have seen, if the particles are electrons, the results are indeed different if we don’t measure the particle between beginning and end points compared to what happens if we do.

Quantum mechanics works, whether or not it makes sense.

I
T IS PRECISELY
this seemingly nonsensical aspect of quantum mechanics that Richard Feynman focused on. As he later put it, the classical picture is wrong if the statement that the position of the particle midway on its trip from
a
to
c
actually takes some specific value,
b
, is wrong. Quantum mechanics instead allows for all possible
paths
, with all values of
b
to be chosen at the same time.